Geometric inequalities for initial data with symmetries

[English]
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Abstract

We consider a class of initial data sets (Σ,h,K) for the Einstein constraint equations
which we define to be generalized Brill (GB) data. This class of data is simply
connected, U(1)²-invariant, maximal, and four-dimensional with two asymptotic ends.
We study the properties of GB data and in particular the topology of Σ. The GB
initial data sets have applications in geometric inequalities in general relativity. We
construct a mass functional M for GB initial data sets and we show:(i) the mass of any
GB data is greater than or equals M, (ii) it is a non-negative functional for a broad
subclass of GB data, (iii) it evaluates to the ADM mass of reduced t − φi symmetric
data set, (iv) its critical points are stationary U(1)²-invariant vacuum solutions to
the Einstein equations. Then we use this mass functional and prove two geometric
inequalities: (1) a positive mass theorem for subclass of GB initial data which includes
Myers-Perry black holes, (2) a class of local mass-angular momenta inequalities for
U(1)²-invariant black holes. Finally, we construct a one-parameter family of initial
data sets which we show can be seen as small deformations of the extreme Myers-
Perry black hole which preserve the horizon geometry and angular momenta but have
strictly greater energy.